We propose and compare goal-oriented projection based model order reduction
methods for the estimation of vector-valued functionals of the solution of
parameter-dependent equations. The first projection method is a generalization
of the classical primal-dual method to the case of vector-valued variables of
interest. We highlight the role played by three reduced spaces: the
approximation space and the test space associated to the primal variable, and
the approximation space associated to the dual variable. Then we propose a
Petrov-Galerkin projection method based on a saddle point problem involving an
approximation space for the primal variable and an approximation space for an
auxiliary variable. A goal-oriented choice of the latter space, defined as the
sum of two spaces, allows us to improve the approximation of the variable of
interest compared to a primal-dual method using the same reduced spaces. Then,
for both approaches, we derive computable error estimates for the
approximations of the variable of interest and we propose greedy algorithms for
the goal-oriented construction of reduced spaces. The performance of the
algorithms are illustrated on numerical examples and compared to standard (non
goal-oriented) algorithms
